Solving Linear Inequalities
This topic covers solving algebraic statements that use 'greater than' or 'less than' instead of 'equals'. For the ESAT, you must be able to manipulate these inequalities and represent the range of possible solutions on a number line for one variable, or as a shaded region on a graph for two variables.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The golden rule of inequalities: if you multiply or divide both sides by a negative number, you MUST reverse the direction of the inequality sign (e.g., `>` becomes `<`).
- For single-variable solutions on a number line, use an open circle (o) for strict inequalities (<, >) and a filled circle (●) for inclusive inequalities (≤, ≥) to mark the endpoint.
- For two-variable inequalities, the solution is a region on a graph. The boundary line is dashed for strict inequalities (<, >) and solid for inclusive ones (≤, ≥).
- To determine which region to shade on a graph, draw the boundary line and then test a coordinate (like (0,0)) in the inequality. If the inequality holds true, shade the region containing that point.
- Compound inequalities, like `-2 < x ≤ 5`, define a single continuous range. They can be manipulated by applying the same operation to all three parts simultaneously.
Diagram
Formulae
If a < b and c < 0, then a*c > b*c This rule applies whenever you multiply or divide both sides of an inequality by a negative number. It is the most critical manipulation rule to remember.
Definitions
- Inequality
- A mathematical statement comparing two expressions that are not necessarily equal, using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
- Solution Set
- The complete set of all values that satisfy an inequality or system of inequalities.
- Boundary Line
- The line that corresponds to the equality in a two-variable inequality. For example, the boundary line for `y > 3x - 1` is `y = 3x - 1`.
Worked example
Find the sum of all integer values of x that satisfy both `4x + 9 > 1` and `10 - 2x ≥ 4`.
- 1
First, solve the inequality `4x + 9 > 1`.
- 2
Subtract 9 from both sides:
`4x > -8`.
- 3
Divide by 4 (a positive number, so the sign stays the same):
`x > -2`.
- 4
Next, solve the second inequality `10 - 2x ≥ 4`.
- 5
Subtract 10 from both sides:
`-2x ≥ -6` - 6
Divide by -2.
Since this is a negative number, you must reverse the inequality sign:
`x ≤ 3` - 7
Now, find the integers that satisfy both conditions:
`x > -2` AND `x ≤ 3`.
- 8
The integers are -1, 0, 1, 2, and 3.
- 9
Finally, calculate their sum:
`(-1) + 0 + 1 + 2 + 3 = 5`.
Answer: 5
Common mistakes
- ×Forgetting to flip the inequality sign when multiplying or dividing by a negative number. For instance, incorrectly simplifying `-5x < 15` to `x < -3` instead of the correct `x > -3`.
- ×When solving a system of inequalities, finding the solution to each one but failing to find the common region or range that satisfies all of them at once.
- ×Using a solid boundary line on a graph for a strict inequality (`<` or `>`), or a dashed line for an inclusive one (`≤` or `≥`). Remember: solid lines include the boundary, dashed lines do not.
- ×Incorrectly shading the graph for a two-variable inequality. Always test a point (like (0,0) if it's not on the line) to confirm which side is the correct solution region.
No-calculator tips
- ✓To quickly solve a compound inequality like `1 ≤ 2x - 5 < 9`, work from the inside out. First add 5 to all three parts to get `6 ≤ 2x < 14`, then divide all parts by 2 to get `3 ≤ x < 7`.
- ✓When faced with two separate inequalities, sketch a simple number line. Draw the range for each inequality above the line. The solution is the segment where your drawn ranges overlap.
- ✓When plotting a boundary line like `2x + 3y = 6`, quickly find the intercepts. Set `x=0` to get `3y=6`, so `y=2`. Set `y=0` to get `2x=6`, so `x=3`. Plot the points (0,2) and (3,0) and connect them.